# Coordinate Geometry Problem Solution
## Problem Statement
Determine the coordinates of the point **P** on the line **y = -x + 6** such that **P** is equidistant from the points **A(10, -10)** and **O(0, 0)** (i.e., **PA = PO**). Express your answer as an ordered pair **(a, b)**.
---
## Solution Process
### Step 1: Represent Point P
Since **P** lies on the line **y = -x + 6**, its coordinates can be expressed as:
$$
P = (x, -x + 6)
$$
### Step 2: Distance Formula
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$
#### Distance from P to A:
$$
PA = \sqrt{(x - 10)^2 + ((-x + 6) + 10)^2}
$$
Simplify:
$$
PA = \sqrt{(x - 10)^2 + (-x + 16)^2}
$$
#### Distance from P to O:
$$
PO = \sqrt{(x - 0)^2 + ((-x + 6) - 0)^2}
$$
Simplify:
$$
PO = \sqrt{x^2 + (-x + 6)^2}
$$
### Step 3: Equate Distances (PA = PO)
$$
\sqrt{(x - 10)^2 + (-x + 16)^2} = \sqrt{x^2 + (-x + 6)^2}
$$
Square both sides to eliminate the square roots:
$$
(x - 10)^2 + (-x + 16)^2 = x^2 + (-x + 6)^2
$$
### Step 4: Expand and Simplify
Expand both sides:
$$
(x^2 - 20x + 100) + (x^2 - 32x + 256) = x^2 + (x^2 - 12x + 36)
$$
Combine like terms:
$$
2x^2 - 52x + 356 = 2x^2 - 12x + 36
$$
Subtract $2x^2$ from both sides:
$$
-52x + 356 = -12x + 36
$$
Solve for $x$:
$$
-40x = -320 \implies x = 8
$$
### Step 5: Find Corresponding $y$-Coordinate
Substitute $x = 8$ into $y = -x + 6$:
$$
y = -8 + 6 = -2
$$
### Final Answer
The coordinates of point **P** are:
$$
\boxed{(8, -2)}
$$
---
## Verification
- **PA**: Distance from $(8, -2)$ to $(10, -10)$:
$$
\sqrt{(8 - 10)^2 + (-2 + 10)^2} = \sqrt{4 + 64} = \sqrt{68}
$$
- **PO**: Distance from $(8, -2)$ to $(0, 0)$:
$$
\sqrt{8^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68}
$$
Both distances are equal, confirming the solution is correct.
